climate change, collapse and social choice theory · czech economic review 9 (2015) 7–35 acta...

Czech Economic Review 9 (2015) 7–35Acta Universitatis Carolinae Oeconomica

Received 1 April 2015; Accepted 1 June 2015

Climate Change, Collapse and Social Choice Theory

Norman Schofield∗

Abstract The enlightenment was a philosophical project to construct a rational society withoutthe need for a supreme being. It opened the way for the creation of market democracy and rapideconomic growth. At the same time economic growth is the underlying cause of climate change,and we have become aware that this may destroy our civilization. The principal underpinningof the enlightenment project is the general equilibrium theorem (GET) of Arrow and Debreu(1954), asserting the existence of a Pareto optimal price equilibrium. Arrow’s work in socialchoice can be interpreted as an attempt to construct a more general social equilibrium theorem.The current paper surveys recent results in social choice which suggests that chaos rather thanequilibrium is generic. We also consider models of belief aggregation similar to Condorcet’sJury Theorem and mention Penn’s theorem on existence of a belief equilibrium. However, it issuggested that a belief equilibrium with regard to the appropriate response to climate change de-pends on the creation of a fundamental social principle of “guardianship of our planetary home.”It is suggested that this will involve conflict between entrenched economic interests and ordinarypeople, as the effects of climate change make themselves felt in many countries.

Keywords The enlightenment, climate change, black swan events, dynamical modelsJEL classification H10 *

1. Introduction

In this essay I shall develop my earlier argument, in Schofield (2011), that human deci-sion making is essentially chaotic. What I mean by this term is that contrary to generalequilibrium theory (Arrow and Debreu 1954) there is no reason, in general, to assumethat we can make wise choices. What Israel (2002) calls the Radical Enlightenment ofthe Eighteenth Century was based on the assumption that it was possible to establisha rational society, opposed to monarchy, religion and the church (see the other worksby Israel 2006, 2010, 2014). Radical enlighteners included Thomas Jefferson, ThomasPaine and James Madison. They believed that society could be based on constitutionalprinciples, leading to the “probability of a fit choice.” Implicit in the Radical Enlighten-ment was the belief, originally postulated by Spinoza, that individuals could find moralbases for their choices without a need for a divine creator. An ancillary belief was thatthe economy would also be rational and that the principles of the Radical Enlighten-ment would lead to material growth and the eradication of poverty and misery. I shallargue here that this enlightenment project has recently had to face two profoundly trou-bling propositions. First are the results of Arrovian social choice theory. These very* Center in Political Economy, Washington University in Saint Louis, 1 Brookings Drive, Saint Louis MO63130, USA. Phone: 3149355630, E-mail: [email protected].

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N. Schofield

abstract results suggest that no process of social choice can be “rational,” in the senseof giving rise to a state of affairs that is acceptable to all. Second, recent events suggestthat the market models that we have used to guide our economic actions are deeplyflawed. In contrast to the Radical enlighteners, both Hume and Burke believed thatpeople would need religion and nationalism to provide a moral compass to their lives.As Putnam and Campbell (2010) have noted, religion today is as important as it hasever been in the U.S. Recent models of U.S. Elections (Schofield and Gallego 2011)show that religion is a key dimension of politics that divides voters one from another.

A consequence of the Industrial Revolution, that followed on from the RadicalEnlightenment, has been the unintended consequence of climate change. Since thisis the most important policy dimension that the world economy currently faces, thispaper will address the question whether we are likely to be able to make wise socialchoices to avoid future catastrophe. To guide us in this, I believe we need a moralcompass founded on religious principles.

1.1 The Radical Enlightenment

A fundamental principle of the enlightenment was the utilization of mathematics touncover nature. Hawking and Mlodinow (2010) argue for a strong version of thisuniversal mathmatical principle, called model-dependent realism, citing the recent de-velopments in mathematical physics and cosmology.

They argue that it is only through a mathematical model that we can properly per-ceive reality. However, this mathematical principle faces two philosophical difficul-ties. One stems from the Godel (1931)-Turing (1937) undecidability theorems. Thefirst theorem asserts that mathematics cannot be both complete and consistent, so thereare mathematical theories that in principle cannot be verified. Turing’s work, though itprovides the basis for our computer technology also suggests that not all programs arecomputable. These two results suggest that our mathematical models of climate andthe economy will be fundamentally uncertain. The second problem is associated withthe notion of chaos or catastrophe.

Since the early work of Garrett Hardin (1968) the “tragedy of the commons” hasbeen recognised as a global prisoners’ dilemma. In such a dilemma no agent has amotivation to provide for the collective good. In the context of the possibility of cli-mate change, the outcome is the continued emission of greenhouses gases like carbondioxide into the atmosphere and the acidification of the oceans. There has developedan extensive literature on the n-person prisoners’ dilemma in an attempt to solve thedilemma by considering mechanisms that would induce cooperation (see for exampleHardin 1971, 1982; Taylor 1976, 1982; Axelrod and Hamilton 1981; Axelrod 1981,1984; Kreps et al. 1982; Margolis 1982).

The problem of cooperation has also provided a rich source of models of evolu-tion, building on the early work by Trivers (1971) and Hamilton (1964, 1970). Nowak(2011) provides an overview of the recent developments. Indeed, the last twenty yearshas seen a growing literature on a game theoretic, or mathematical, analysis of theevolution of social norms to maintain cooperation in prisoners’ dilemma like situa-tions. Gintis (2000), for example, provides evolutionary models of the cooperation

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through strong reciprocity and internalization of social norms.1 The anthropologicalliterature provides much evidence that, from about 500,000 years ago, the ancestorsof homo sapiens engaged in cooperative behavior, particularly in hunting and caringfor offspring and the elderly.2 On this basis we can infer that we probably do havevery deeply ingrained normative mechanisms that were crucial, far back in time, forthe maintenance of cooperation, and the fitness and thus survival of early hominids.3

These normative systems will surely have been modified over the long span of ourevolution.

Current work on climate change has focussed on how we should treat the future.For example Stern (2007, 2009), Collier (2010) and Chichilnisky (2009a,b) argue es-sentially for equal treatment of the present and the future. Dasgupta (2005) points outthat how we treat the future depends on our current estimates of economic growth inthe near future.

The fundamental problem of climate change is that the underlying dynamic systemis extremely complex, and displays many positive feedback mechanisms (see the dis-cussion in Schofield 2011). The difficulty can perhaps be illustrated by Figure 1. It isusual in economic analysis to focus on Pareto optimality. Typically in economic the-ory, it is assumed that preferences and production possibilities are generated by convexsets. However, climate change could create non-convexities. In such a case the Paretoset will exhibit stable and unstable components. Figure 1 distinguishes between a do-main A, bounded by stable and unstable components Ps

1 and Pu, and a second stablecomponent Ps

2 . If our actions lead us to an outcome within A, whether or not it is Pare-tian, then it is possible that the dynamic system generated by climate could lead to acatastrophic destruction of A itself. More to the point, our society would be trappedinside A as the stable and unstable components merged together.

Our society has recently passed through a period of economic disorder, where“black swan” events, low probability occurrences with high costs, have occurred withsome regularity. Recent discussion of climate change has also emphasized so called“fat-tailed climate events” again defined by high uncertainty and cost (Weitzman 2009;Chichilnisky 2010).4 The catastrophic change implied by Figure 1 is just such a blackswan event. The point to note about Figure 1 is everything would appear normal untilthe evaporation of A.

Cooperation could in principle be attained by the action of a hegemonic leader suchas the United States as suggested by Kindleberger (1973) and Keohane and Nye (1977).In Section 2 we give a brief exposition of the prisoners’ dilemma and illustrate howhegemonic behavior could facilitate international cooperation. However, the analysissuggests that in the present economic climate, such hegemonic leadership is unlikely.

1 Strong reciprocity means the punishment of those who do not cooperate.2 Indeed, White et al. (2009) present evidence of a high degree of cooperation among very early hominidsdating back about 4MYBP (million years before the present). The evidence includes anatomical data whichallows for inferences about the behavioral characteristics of these early hominids.3 Gintis cites the work of Robson and Kaplan (2003) who use an economic model to estimate the correlationbetween brain size and life expectancy (a measure of efficiency). In this context, the increase in brain size isdriven by the requirement to solve complex cooperative games against nature.4 See also Chichilnisky and Eisenberger (2010) on other catastophic events such as collision with an asteroid.


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Ps2

Ps1

PuA

Figure 1. Stable and unstable components of the global Pareto set

Analysis of games such as the prisoner’s dilemma usually focus on the existenceof a Nash equilibrium, a vector of strategies with the property that no agent has anincentive to change strategy. Section 3 considers the family of equilibrium modelsbased on the Brouwer (1912) fixed point theorem, or the more general result known asthe Ky Fan theorem (Fan 1961) as well as the application by Bergstrom (1975, 1992)to prove existence of a Nash equilibrium and market equilibrium.

Section 4 considers a generalization of the Ky Fan Theorem, and argues that thegeneral equilibrium argument can be interpreted in terms of particular properties of apreference field, H, defined on the tangent space of the joint strategy space. If thisfield is continuous, in a certain well-defined sense, and “half-open” then it will exhibita equilibrium. This half-open property is the same as the non-empty intersection of afamily of dual cones. We mention a theorem by Chichilnisky (1995) that a necessaryand sufficient condition for market equilibrium is that a family of dual cones also hasnon-empty intersection.

However, preference fields that are defined in terms of coalitions need not satisfythe half-open property and thus need not exhibit equilibrium. For coalition systems,it can be shown that unless there is a collegium or oligarchy, or the dimension of thespace is restricted in a particular fashion, then there need be no equilibrium. Earlierresults by McKelvey (1976), Schofield (1978), McKelvey and Schofield (1987) andSaari (1997) suggested that voting can be “non-equilibriating” and indeed “chaotic”(see Schofield 1977, 1980a,b).5

Kauffman (1993) commented on “chaos” or the failure of “structural stability” inthe following way.

“One implication of the occurrence or non-occurrence of structural sta-bility is that, in structurally stable systems, smooth walks in parameter

5 In a sense these voting theorems can be regarded as derivative of Arrow’s Impossiblity Theorem (Arrow1951). See also Arrow (1986).



space must [result in] smooth changes in dynamical behavior. By con-trast, chaotic systems, which are not structurally stable, adapt on uncorre-lated landscapes. Very small changes in the parameters pass through manyinterlaced bifurcation surfaces and so change the behavior of the systemdramatically.”

Chaos is generally understood as sensitive dependence on initial conditions whereasstructural stability means that the qualitative nature of the dynamical system does notchange as a result of a small perturbation.6 I shall use the term chaos to mean that thetrajectory taken by the dynamical process can wander anywhere.7

An earlier prophet of uncertainty was, of course, Keynes (1936) whose ideas on“speculative euphoria and crashes” would seem to be based on understanding the eco-nomy in terms of the qualitative aspects of its coalition dynamics (see Minsky 1975,1986 and Keynes’s earlier work in 1921). An extensive literature has tried to draw in-ferences from the nature of the recent economic events. A plausible account of marketdisequilibrium is given by Akerlof and Shiller (2009) who argue that

“. . . the business cycle is tied to feedback loops involving speculative pricemovements and other economic activity—and to the talk that these move-ments incite. A downward movement in stock prices, for example, gen-erates chatter and media response, and reminds people of longstandingpessimistic stories and theories. These stories, newly prominent in theirminds, incline them toward gloomy intuitive assessments. As a result, thedownward spiral can continue: declining prices cause the stories to spread,causing still more price declines and further reinforcement of the stories.”

It would seem reasonable that the rise and fall of the market is due precisely tothe coalitional nature of decision-making, as large sets of agents follow each other inexpecting first good things and then bad. A recent example can be seen in the fall in themarket after the earthquake in Japan, and then recovery as an increasing set of investorsgradually came to believe that the disaster was not quite as bad as initially feared.

Since investment decisions are based on these uncertain evaluations, and these arethe driving force of an advanced economy, the flow of the market can exhibit singular-ities, of the kind that recently nearly brought on a great depression. These singularitiesassociated with the bursting of market bubbles are time-dependent, and can be inducedby endogenous belief-cascades, rather than by any change in economic or politicalfundamentals (Corcos et al. 2002).

Similar uncertainty holds over political events. The fall of the Berlin Wall in 1989was not at all foreseen. Political scientists wrote about it in terms of “belief cascades”(Karklins and Petersen 1993; Lohmann 1994; see also Bikhchandani et al. 1992) as6 The theory of chaos or complexity is rooted in Smale’s fundamental theorem (Smale 1966) that struc-tural stability of dynamical systems is not “generic” or typical whenever the state space has more than twodimensions.7 In their early analysis of chaos, Li and Yorke (1975) showed that in the domain of a chaotic transformationf it was possible for almost any pair of positions (x,y) to transition from x to y = f r(x), where f r means ther times reiteration of f .


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the coalition of protesting citizens grew apace. As the very recent democratic revo-lutions in the Middle East and North Africa suggest, these coalitional movements areextremely uncertain.8 In particular, whether the autocrat remains in power or is forcedinto exile is as uncertain as anything Keynes discussed. Even when democracy isbrought about, it is still uncertain whether it will persist (see for example Carothers2002 and Collier 2009).

Section 5 introduces the Condorcet (1994 [1795]) Jury Theorem. This theoremsuggests that majority rule can provide a way for a society to attain the truth whenthe individuals have common goals. Schofield (2002, 2006) has argued that Madisonwas aware of this theorem while writing Federalist X (Madison 1999 [1787]) so itcan be taken as perhaps the ultimate justification for democracy. However, models ofbelief aggregation that are derived from the Jury Theorem can lead to belief cascadesthat bifurcate the population. In addition, if the aggregation process takes place ona network, then centrally located agents, who have false beliefs, can dominate theprocess (Golub and Jackson 2010).

In Section 6 we introduce the idea of a belief equilibrium, and then go on to con-sider the notion of “punctuated equilibrium” in general evolutionary models. Againhowever, the existence of an equilibrium depends on a fixed point argument, and thuson a half-open property of the “cones” by which the developmental path is modeled.This half-open property is equivalent to the existence of a social direction gradient de-fined everywhere. In Section 7 we introduce the notion of a “moral compass” that mayprovide a teleology to guide us in making wise choices for the future, by providing uswith a social direction gradient. Section 8 concludes.

2. The Prisoners’ Dilemma, Cooperation and Morality

“For before constitution of Sovereign Power . . . all men had right to allthings; which necessarily causeth Warre.” (Hobbes 2009 [1651])

Kindleberger (1973) gave the first interpretation of the international economic sys-tem of states as a “Hobbesian” prisoners’ dilemma, which could be solved by a leader,or “hegemon.”

“A symmetric system with rules for counterbalancing, such as the goldstandard is supposed to provide, may give way to a system with each par-ticipant seeking to maximize its short-term gain . . . But a world of a fewactors (countries) is not like [the competitive system envisaged by AdamSmith] . . . In advancing its own economic good by a tariff, currency de-preciation, or foreign exchange control, a country may worsen the welfareof its partners by more than its gain. Beggar-thy-neighbor tactics may leadto retaliation so that each country ends up in a worse position from havingpursued its own gain . . . This is a typical non-zero sum game, in which any

8 The response by the citizens of these countries to the demise of Osama bin Laden on May 2, 2011, is inlarge degree also unpredictable.



player undertaking to adopt a long range solution by itself will find othercountries taking advantage of it . . . ”

In the 1970s, Robert Keohane and Joseph Nye (1977) rejected “realist” theory ininternational politics, and made use of the idea of a hegemonic power in a context of“complex interdependence” of the kind envisaged by Kindleberger. Although they didnot refer to the formalism of the prisoners’ dilemma, it would appear that this notiondoes capture elements of complex interdependence. To some extent, their concept ofa hegemon is taken from realist theory rather than deriving from the game-theoreticformalism.

The essence of the theory of hegemony in international relations is that if there isa degree of inequality in the strengths of nation states then a hegemonic power maymaintain cooperation in the context of an n-country prisoners’ dilemma. Clearly, theBritish Empire in the 1800’s is the role model for such a hegemon (Ferguson 2002).

Hegemon theory suggests that international cooperation was maintained after WorldWar II because of a dominant cooperative coalition. At the core of this cooperativecoalition was the United States; through its size it was able to generate collective goodsfor this community, first of all through the Marshall Plan and then in the context first ofthe post-world war II system of trade and economic cooperation, based on the BrettonWoods agreement and the Atlantic Alliance, or NATO. Over time, the United Stateshas found it costly to be the dominant core of the coalition, in particular, as the relativesize of the U.S. economy has declined. Indeed, the global recession of 2008–2010suggests that problems of debt could induce “begger thy neighbor strategies,” just likethe 1930’s.

The future utility benefits of adopting policies to ameliorate these possible changesdepend on the discount rates that we assign to the future. Dasgupta (2005) gives a clearexposition of how we might assign these discount rates. Obviously enough, differentcountries will in all likelihood adopt very different evaluations of the future. Develop-ing countries like the BRICs (Brazil, Russia, India and China) will choose growth anddevelopment now rather than choosing consumption in the future.

An extensive literature over the last few years has developed Adam Smith’s ideas asexpressed in the Theory of Moral Sentiments (1984 [1759]) to argue that human beingshave an innate propensity to cooperate. This propensity may well have been the resultof co-evolution of language and culture (Boyd and Richerson 2005; Gintis 2000).

Since language evolves very quickly (McWhorter 2001; Deutcher 2006), we mightalso expect moral values to change fairly rapidly, at least in the period during whichlanguage itself was evolving. In fact there is empirical evidence that cooperative beha-vior as well as notions of fairness vary significantly across different societies.9 Whilethere may be fundamental aspects of morality and “altruism,” in particular, held incommon across many societies, there is variation in how these are articulated. Gaz-zaniga (2008) suggests that moral values can be described in terms of various modules:reciprocity, suffering (or empathy), hierarchy, in-group and outgroup coalition, and

9 See Henrich et al. (2004, 2005), which reports on experiments in fifteen “small-scale societies,” using thegame theoretic tools of the “prisoners’ dilemma,” the “ultimatum game,” etc.


N. Schofield

purity/disgust. These modules can be combined in different ways with different em-phases. An important aspect of cooperation is emphasized by Burkart et al. (2009)and Hrdy (2011), namely cooperation between man and woman to share the burden ofchild rearing.

It is generally considered that hunter-gatherer societies adopted egalitarian or “fairshare” norms. The development of agriculture and then cities led to new norms of hi-erarchy and obedience, coupled with the predominance of military and religious elites(Schofield 2010).

North (1990), North et al. (2009) and Acemoglu and Robinson (2006) focus onthe transition from such oligarchic societies to open access societies whose institu-tions or “rules of the game,” protect private property, and maintain the rule of law andpolitical accountability, thus facilitating both cooperation and economic development.Acemoglu et al. (2009) argue, in their historical analyses about why “good” institu-tions form, that the evidence is in favor of “critical junctures” (see also Acemoglu andRobinson 2008). For example, the “Glorious Revolution” in Britain in 1688 (North andWeingast 1989), which prepared the way in a sense for the agricultural and industrialrevolutions to follow (Mokyr 2005, 2010; Mokyr and Nye 2007) was the result of a se-quence of historical contingencies that reduced the power of the elite to resist change.Recent work by Morris (2010), Fukuyama (2011), Ferguson (2011) and Acemoglu andRobinson (2011) has suggested that these fortuitous circumstances never occurred inChina and the Middle East, and as a result these domains fell behind the West. Al-though many states have become democratic in the last few decades, oligarchic poweris still entrenched in many parts of the world.10

At the international level, the institutions that do exist and that are designed tomaintain cooperation, are relatively young. Whether they succeed in facilitating coop-eration in such a difficult area as climate change is a matter of speculation. As we havesuggested, international cooperation after World War II was only possible because ofthe overwhelming power of the United States. In a world with oligarchies in powerin Russia, China, and in many countries in Africa, together with political disorder inalmost all the oil producing counties in the Middle East, cooperation would appearunlikely.

To extend the discussion, we now consider more general theories of social choice.

3. Existence of a Choice

The above discussion has considered a very simple version of the prisoner’s dilemma.The more general models of cooperation typically use variants of evolutionary gametheory, and in essence depend on proof of existence of Nash equilibrium, using someversion of the Brouwer’s fixed point theorem (Brouwer 1912).

Brouwer’s theorem asserts that any continuous function f : B → B from the finitedimensional ball, B (or indeed any compact convex set in Rw) into itself, has the fixedpoint property. That is, there exists some x ∈ B such that f (x) = x.

10 The popular protests in North Africa and the Middle East in 2011 were in opposition to oligarchic andautocratic power.



We will now consider the use of variants of the theorem, to prove existence ofan equilibrium of a general choice mechanism. We shall argue that the condition forexistence of an equilibrium will be violated if there are cycles in the underlying me-chanism.

Let W be the set of alternatives and let X be the set of all subsets of W. A preferencecorrespondence, P, on W assigns to each point x ∈ W, its preferred set P(x). WriteP : W → X or P : W � W to denote that the image of x under P is a set (possiblyempty) in W. For any subset V of W, the restriction of P to V gives a correspondencePV : V �V. Define P−1

V : V �V such that for each x ∈V,

P−1V (x) = {y : x ∈ P(y)}∩V.

The sets PV (x),P−1V (x) are sometimes called the upper and lower preference sets of P

on V. When there is no ambiguity we delete the suffix V. The choice of P from W isthe set

C(W,P) = {x ∈W : P(x) =∅} .

Here ∅ is the empty set. The choice of P from a subset, V, of W is the set

C(V,P) = {x ∈V : PV (x) =∅} .

Call CP a choice function on W if CP(V ) = C(V,P) 6= ∅ for every subset V of W.We now seek general conditions on W and P which are sufficient for CP to be a choicefunction on W. Continuity properties of the preference correspondence are importantand so we require the set of alternatives to be a topological space.

Definition 1. Let W,Y be two topological spaces. A correspondence P : W � Y is

(i) Lower demi-continuous (ldc) iff, for all x ∈ Y, the set

P−1 (x) = {y ∈W : x ∈ P(y)}

is open (or empty) in W .

(ii) Acyclic if it is impossible to find a cycle xt ∈ P(xt−1), xt−1 ∈ P(xt−2), . . . , x1 ∈P(xt).

(iii) Lower hemi-continuous (lhc) iff, for all x ∈ W, and any open set U ⊂ Y suchthat P(x)∩U 6= ∅ there exists an open neighborhood V of x in W, such thatP(x′)∩U 6=∅ for all x′ ∈V.

Note that if P is ldc then it is lhc.

We shall use lower demi-continuity of a preference correspondence to prove exis-tence of a choice.

We shall now show that if W is compact, and P is an acyclic and ldc preferencecorrespondence P : W � W, then C(W,P) 6= ∅. First of all, say a preference corre-spondence P : W �W satisfies the finite maximality property (FMP) on W iff for everyfinite set V in W, there exists x ∈V such that P(x)∩V =∅.


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Lemma 1. (Walker 1977) If W is a compact, topological space and P is an ldc prefe-rence correspondence that satisfies FMP on W, then C(W,P) 6=∅.

This follows readily, using compactness to find a finite subcover, and then using FMP.

Corollary 1. If W is a compact topological space and P is an acyclic, ldc preferencecorrespondence on W, then C(W,P) 6=∅.

As Walker (1977) noted, when W is compact and P is ldc, then P is acyclic iff Psatisfies FMP on W, and so either property can be used to show existence of a choice. Asecond method of proof is to show that CP is a choice function to substitute a convexityproperty for P rather than acyclicity.

Definition 2.

(i) If W is a subset of a vector space, then the convex hull of W is the set, Con[W ],defined by taking all convex combinations of points in W.

(ii) W is convex iff W =Con[W ]. (The empty set is also convex.)

(iii) W is admissible iff W is a compact, convex subset of a topological vector space.

(iv) A preference correspondence P : W � W is semi-convex iff, for all x ∈ W, it isthe case that x /∈Con(P(x)).

Fan (1961) has shown that if W is admissible and P is ldc and semi-convex, thenC(W,P) is non-empty.

Theorem 1 (Choice Theorem). (Fan 1961, Bergstrom 1975) If W is an admissiblesubset of a Hausdorff topological vector space, and P : W � W a preference corre-spondence on W which is ldc and semi-convex then C(W,P) 6=∅.

The proof uses the KKM lemma due to Knaster, Kuratowski and Mazurkiewicz (1929).The original form of the Fan Theorem made the assumption that P : W � W was

irreflexive (in the sense that x /∈ P(x) for all x ∈ W ) and convex. Together these twoassumptions imply that P is semi-convex. Bergstrom (1975) extended Fan’s originalresult to give the version presented above (see also Shafer and Sonnenschein 1975).

Note that the Fan Theorem is valid without restriction on the dimension of W. In-deed, Aliprantis and Brown (1983) have used this theorem in an economic context withan infinite number of commodities to show existence of a price equilibrium. Bergstrom(1992) also showed that when W is finite dimensional then the Fan Theorem is validwhen the continuity property on P is weakened to lhc and used this theorem to show ex-istence of a Nash equilibrium of a game G= {(P1,W1), . . . ,Pi,Wi), . . . ,(Pn,Wn) : i∈N}.Here the ith stategy space is finite dimensional Wi and each individual has a preferencePi on the joint strategy space Pi : W N = W1 ×W2 . . .×Wn � Wi. The Fan Theoremcan be used, in principle to show existence of an equilibrium in complex economieswith externalities. Define the Nash improvement correspondence by P∗

i : W N � W N

by y ∈ P∗i (x) whenever y = (x1, . . . ,xi−1,x∗i , . . . ,xn), x = (x1, . . . ,xi−1,xi, . . . ,xn), and

x∗i ∈ Pi(x). The joint Nash improvement correspondence is P∗N = ∪P∗

i : W N � W N .The Nash equilibrium of a game G is a vector z ∈ W N such that P∗

N(z) =∅. Then theNash equilibrium will exist when P∗

N is ldc and semi-convex and W N is admissible.



4. Dynamical Choice Functions

We now consider a generalized preference field H : W � TW, on a manifold W. TWis the tangent bundle above W, given by TW = ∪{TxW : x ∈ W}, where TxW is thetangent space above x. If V is a neighborhood of x, then TVW = ∪{TxW : x ∈V} whichis locally like the product space Rw ×V. Here W is locally like Rw.

At any x ∈W, H(x) is a cone in the tangent space TxW above x. That is, if a vectorv ∈ H(x), then λv ∈ H(x) for any λ > 0. If there is a smooth curve, c : [−1,1]→ W,

such that the differential dc(t)dt ∈ H(x), whenever c(t) = x, then c is called an integral

curve of H. An integral curve of H from x = c(o) to y = limt→1 c(t) is called an H-preference curve from x to y. In this case we write y∈H(x). We say y is reachable fromx if there is a piecewise differentiable H−preference curve from x to y, so y ∈Hr(x) forsome reiteration r. The preference field H is called S-continuous iff the inverse relationH−1 is ldc. That is, if x is reachable from y, then there is a neighborhood V of y suchthat x is reachable from all of V. The choice C(W,H) of H on W is defined by

C(W,H) = {x ∈W : H(x) =∅}.

Say H(x) is semi-convex at x ∈ W, if either H(x) = ∅ or 0 /∈ Con[H(x)] in thetangent space TxW . In the later case, there will exist a vector v′ ∈ TxW such that (v′ ·v)>0 for all v∈H(x). We can say in this case that there is, at x, a direction gradient d in thecotangent space T ∗

x W of linear maps from TxW to R such that d(v)> 0 for all v∈H(x).If H is S-continuous and half-open in a neighborhood, V, then there will exist such acontinuous direction gradient d : V → T ∗V on the neighborhood V .11

We define

Cycle(W,H) = {x ∈W : H(x) 6=∅,0 ∈ConH(x)}.

Definition 3. The dual of a preference field H : W � TW is defined by H∗ : W �T ∗W : x � {d ∈ T ∗

x W : d(v)> 0 for all v ∈ H(x)⊂ TxW}. For convenience if H(x) =∅we let H∗(x) = TxW. Note that if 0 /∈ConH(x) iff H∗(x) 6=∅. We can say in this casethat the field is half-open at x.

In applications, the field H(x) at x will often consist of some family {H j(x)}. Asan example, let u : W � Rn be a smooth utility profile and for any coalition M ⊂ N let

HM(u)(x) = {v ∈ TxW : dui(x)(v)> 0, ∀i ∈ M}.

If D is a family of decisive coalitions, D= {M ⊂ N}, then we define

HD(u) = ∪HM(u) : W � TW.

Then the field HD(u) :W �TW has a dual [HD(u)]∗ :W �T ∗W given by [HD(u)]

∗(x)=∩[HM(u)(x)]∗ where the intersection at x is taken over all M ∈D such that HM(u)(x) 6=∅. We call [HM(u)(x)]∗ the co-cone of [HM(u)(x)]∗. It then follows that at x ∈11 i.e. d(x)(v)> 0 for all x ∈V, for all v ∈ H(x), whenever H(x) 6=∅.


N. Schofield

Cycle(W,HD(u)) then 0 ∈Con[HD(u)(x)] and so [HD(u)(x)]∗ =∅. Thus

Cycle(W,HD(u)) = {x ∈W : [HD(u)]∗(x) =∅}.

The condition that [HD(u)]∗(x) =∅ is equivalent to the condition ∩[HM(u)(x)]∗ =

∅ and was called the null dual condition (at x). Schofield (1978) has shown thatCycle(W,HD(u)) will be an open set and contains cycles so that a point x is reachablefrom itself through a sequence of preference curves associated with different coalitions.This result was an application of a more general result.

Theorem 2 (Dynamical Choice Theorem). (Schofield 1978) For any S-continuousfield H on compact, convex W, then

Cycle(W,H)∪C(W,H) 6=∅.

Proof. If x ∈ Cycle(W,H) 6= ∅ then there is a piecewise differentiable H-preferencecycle from x to itself. If there is an open path connected neighborhood V ⊂Cycle(W,H)such that H(x′) is open for all x′ ∈V then there is a piecewise differentiable H-preferencecurve from x to x′. �

(Here piecewise differentiable means the curve is continuous, and also differen-tiable except at a finite number of points.) The proof follows from the previous choicetheorem. The trajectory is built up from a set of vectors {v1, . . . ,vt} each belonging toH(x) with 0 ∈ Con[{v1, . . . ,vt}]. If H(x) is of full dimension, as in the case of a vot-ing rule, then just as in the model of chaos by Li and York(1975), trajectories definedin terms of H can wander anywhere within any open path connected component ofCycle(W,H). Chichilnisky (1997a) has obtained an analogous result.

Theorem 3 (Chichilnisky Theorem). (Chichilnisky 1997a) The limited arbitragecondition ∩[Hi(u)]∗ 6=∅ is necessary and sufficient for existence of a competitive equi-librium.

Chichilnisky (1993, 1997b) also defined a topological obstruction to the non-empti-ness of this intersection and showed the connection with the existence of a social choiceequilibrium.

For a voting rule, D it is possible to guarantee that Cycle(W,HD) =∅ and thus thatC(W,HD) 6=∅. We can do this by restricting the dimension of W.

Extending the equilibrium result of the Nakamura Theorem (Nakamura 1979) tohigher dimension for a voting rule faces a difficulty caused by Bank’s Theorem. Wefirst define a fine topology on smooth utility functions (Hirsch 1976; Schofield 1999,2003).

Definition 4. Let (U(W )N ,T1) be the topological space of smooth utility profiles en-dowed with the the C1-topology.

In economic theory, the existence of isolated price equilibria can be shown to be“generic” in this topological space (Debreu 1970, 1976; Smale 1974a,b). In socialchoice no such equilibrium theorem holds. The difference is essentially because of thecoalitional nature of social choice.



Theorem 4 (Banks Theorem). For any non-collegial D, there exists an integer w(D)such that dim(W )> w(D) implies that C(W,HD(u)) =∅ for all u in a dense subspaceof (U(W )N ,T1) so Cycle(W,HD(u)) 6=∅ generically.

Proof. This result was essentially proved by Banks (1995), building on earlier re-sults by Plott (1967), Kramer (1973), McKelvey (1979), Schofield (1983a,b), McKel-vey and Schofield (1987). See for related analyses. Indeed, it can be shown that ifdim(W ) > w(D)+1 then Cycle(W,HD(u)) is generically dense (Schofield 1984). Theinteger w(D) can usually be computed explicitly from D. For majority rule with n oddit is known that w(D) = 2 while for n even, w(D) = 3. �

Although the Banks Theorem formally applies only to voting rules, Schofield (2010)argues that it is applicable to any non-collegial social mechanism, say H(u) and can beinterpreted to imply that

Cycle(W,H(u)) 6=∅ and C(W,H(u)) =∅

is a generic phenomenon in coalitional systems. Because preference curves can wan-der anywhere in any open component of Cycle(W,H(u)), Schofield (1979) called thischaos. It is not so much the sensitive dependence on initial conditions, but the aspectof indeterminacy that is emphasized. On the other hand, existence of a hegemon, asdiscussed in Section 2, suggests that Cycle(W,H(u)) would be constrained in this case.

Richards (1990) has examined data on the distribution of power in the internationalsystem over the long run and and presents evidence that it can be interpreted in terms ofa chaotic trajectory. This suggests that the metaphor of the nPD in international affairsdoes characterise the ebb and flow of the system and the rise and decline of hegemony.

It is worth noting that the early versions of the Banks Theorem were obtained inthe decade of the 1970’s, a decade that saw the first oil crisis, the collapse of theBretton Woods system of international political economy, the apparent collapse of theBritish economy, the beginning of social unrest in Eastern Europe, the revolution inIran, and and the second oil-crisis (Caryl 2011). Many of the transformations that haveoccurred since then can be seen as changes in beliefs, rather than preferences. Modelsof belief aggregation are less well developed than those dealing with preferences.12 Ingeneral models of belief aggregation are related to what is now termed Condorcet’sJury Theorem, which we now introduce.

5. Beliefs and Condorcet’s Jury Theorem

The Jury Theorem formally only refers to a situation where there are just two alterna-tives {1,0}, and alternative 1 is the “true” option. Further, for every individual, i, itis the case that the probability that i picks the truth is ρi1, which exceeds the probabi-lity, ρi0, that i does not pick the truth. We can assume that ρi1 +ρi0 = 1, so obviouslyρi1 >

12 . To simplify the proof, we can assume that ρi1 is the same for every individual,

thus ρi1 = α > 12 for all i. We use χi(= 0 or 1) to refer to the choice of individual i,

12 Results on belief aggregation include Penn (2009) and McKelvey and Page (1986).


N. Schofield

and let χ = Σni=1χi be the number of individuals who select the true option 1. We use

Pr for the probability operator, and E for the expectation operator. In the case that theelectoral size, n, is odd, then a majority, m, is defined to be m = n+1

2 . In the case n iseven, the majority is m = n

2 +1. The probability that a majority chooses the true optionis then

αnma j = Pr[χ ≥ m].

The theorem assumes that voter choice is pairwise independent, so that Pr(χ = j)is simply given by the binomial expression

nj

α j(1−α)n− j.

A version of the theorem can be proved in the case that the probabilities {ρi1 = αi}differ but satisfy the requirement that 1

n Σni=1αi >

12 . Versions of the theorem are valid

when voter choices are not pairwise independent (Ladha and Miller 1996).

Theorem 5 (The Jury Theorem). If 1 > α > 12 , then αn

ma j ≥ α, and αnma j → 1 as

n → ∞.

Proof. For both n being even or odd, as n → ∞, the fraction of voters choosing option1 approaches 1

n E(χ) = α > 12 . Thus, in the limit, more than half the voters choose the

true option. Hence the probability αnma j → 1 as n → ∞. �

Laplace also wrote on the topic of the probability of an error in the judgementof a tribunal. He was concerned with the degree to which jurors would make justdecisions in a situation of asymmetric costs, where finding an innocent party guiltywas to be more feared than letting the guilty party go free. As he commented on theappropriate rule for a jury of twelve, “I think that in order to give a sufficient guaranteeto innocence, one ought to demand at least a plurality of nine votes in twelve” (Laplace1951[1814], p. 139). Schofield (1972a,b) considered a model derived from the JuryTheorem where uncertain citizens were concerned to choose an ethical rule whichwould minimize their disappointment over the the likely outcomes. He showed thatmajority rule was indeed optimal in this sense.

Models of belief aggregation extend the Jury Theorem by considering a situationwhere individuals receive signals, update their beliefs and make an aggregate choiceon the basis of their posterior beliefs (Austen-Smith and Banks 1996). Models of thiskind can be used as the basis for analysing correlated beliefs.13 and the creation ofbelief cascades (Easley and Kleinberg 2010).

Schofield (2002, 2006) has argued that Condorcet’s Jury Theorem provided the ba-sis for Madison’s argument in Federalist X (Madison 1999 [1787]) that the judgmentsof citizens in the extended Republic would enhance the “probability of a fit choice.”However, Schofield’s discussion suggests that belief cascades can also fracture the so-ciety in two opposed factions, as in the lead up to the Civil War in 1860.14

There has been a very extensive literature recently on cascades (Gleick 1987; Bu-chanan 2001, 2003; Gladwell 2002; Johnson 2002; Barabasi 2003, 2010; Strogatz2004; Watts 2002, 2003; Surowiecki 2005; Ball 2004; Christakis and Fowler 2011),

13 Schofield 1972 a,b; Ladha 1992, 1993; Ladha and Miller 1996.14 Sunstein (2006, 2011) also notes that belief aggregation can lead to a situation where subgroups in thesociety come to hold very disparate opinions.



but it is unclear from this literature whether cascades will be equilibriating or veryvolatile. In their formal analysis of cascades on a network of social connections, Goluband Jackson (2010) use the term wise if the process can attain the truth. In particularthey note that if one agent in the network is highly connected, then untrue beliefs of thisagent can steer the crowd away from the truth. The recent economic disaster has led toresearch on market behavior to see if the notion of cascades can be used to explain whymarkets can become volatile or even irrational in some sense (Acemoglu et al. 2010;Schweitzer et al. 2009). Indeed the literature that has developed in the last few yearshas dealt with the nature of herd instinct, the way markets respond to speculative beha-vior and the power law that characterizes market price movements (see, for example,Mandelbrot and Hudson 2004; Shiller 2003, 2005; Taleb 2007; Barbera 2009; Cassidy2009; Fox 2009). The general idea is that the market can no longer be regarded asefficient. Indeed, as suggested by Ormerod (2001) the market may be fundamentallychaotic.

“Empirical” chaos was probably first discovered by Lorenz (1962, 1963) in hisefforts to numerically solve a system of equations representative of the behavior ofweather. A very simple version is the non-linear vector equation

dxdt

=

dx1dx2dx3

=

−a1(x1 − x2)−x1x3 + a2x1 − x2

x1x2 −a3x3

,

which is chaotic for certain ranges of the three constants, a1,a2,a3.The resulting “butterfly” portrait winds a number of times about the left hole (as in

Figure 2), then about the right hole, then the left, etc. Thus the “phase portrait” of thisdynamical system can be described by a sequence of winding numbers (w1

l ,w1k ,w

2l ,w

2k ,

etc.). Changing the constants a1,a2,a3 slightly changes the winding numbers. Notethat Figure 2 is in three dimensions. The butterfly wings on left and right consist ofinfinitely many closed loops. An illustration of the butterfly is the chaotic trajectoryof the Artemis Earth Moon orbiter (which can be found at nasa.gov-artemis orbiter).The butterfly is also called the Lorentz “strange attractor.”A slight perturbation of thisdynamic system changes the winding numbers and thus the qualitative nature of theprocess. Clearly this dynamic system is not structurally stable, in the sense used byKauffmann (1993). The metaphor of the butterfly gives us pause, since all dynamicsystems whether models of climate, markets, voting processes or cascades may beindeterminate or chaotic.

6. The Edge of Chaos

A dynamic belief equilibrium at τ for a society Nτ is a fixed point of a transformation inthe beliefs of the society. Although the space will be infinite dimensional, if the domainand range of this transformation are restricted to equicontinous functions (Pugh 2002),then the domain and range will be compact. Penn (2009) shows that if the domainand range are convex then a generalized version of Brouwer’s fixed point theorem canbe applied to show existence of such a dynamic belief equilibrium. This notion of


N. Schofield

Figure 2. The butterfly

equilibrium was first suggested by Hahn (1973) who argued that equilibrium is locatedin the mind, not in behavior.

However, the choice theorem suggests that the validity of Penn’s result will de-pend on how the model of social choice is constructed. For example Corcos et al.(2002) consider a formal model of the market, based on the reasoning behind Keynes’s“beauty contest” (Keynes 1936). There are two coalitions of “bulls” and “bears.” In-dividuals randomly sample opinion from the coalitions and use a critical cutoff-rule.For example if the individual is bullish and the sampled ratio of bears exceeds someproportion then the individual flips to bearish. The model is very like that of the JuryTheorem but instead of guaranteeing a good choice the model can generate chaoticflips between bullish and bearish markets, as well as fixed points or cyclic behavior,depending on the cut-off parameters. Taleb’s argument (Taleb 2007) about black swanevents can be applied to the recent transformation in societies in the Middle East andNorth Africa that resemble such a cascade (Taleb and Blyth 2011). As in the earlierepisodes in Eastern Europe, it would seem plausible that the sudden onset of a cascadeis due to a switch in a critical coalition.

The notion of “criticality” has spawned in enormous literature particularly in fieldsinvolving evolution, in biology, language and culture (see for example Cavallli-Sforzaand Feldman 1981; Bowles et al. 2003). Bak and Sneppen (1993) refer to the selforganized critical state as the

“. . . ‘edge of chaos’ since it separates a frozen inactive state from a ‘hot’disordered state . . . The mechanism of evolution in the critical state can bethought of as an exploratory search for better local fitness, which is rarelysuccessful, but sometimes has enormous effect on the ecosystem.”

Flyvbjerg et al. (1993) go on to say

“. . . species sit at local fitness maxima . . . and occasionally a species jumpsto another maximum [in doing so it] may change the fitness landscapes ofother species which depend on it . . . Consequently they immediately jump



du1(x)du3(x)

du2(x)

a

b

c x

I1 I3

I2

Figure 3. Cycles in a neighborhood of x

to new maxima. This may affect yet another species in a chain reaction, aburst of evolutionary activity.”

This work was triggered by the earlier ideas on “punctuated equilibrium” by El-dredge and Gould (1972) (see also Eldredge 1976; Gould 1976).

The point to be emphasized is that the evolution of a species involves bifurcation,a splitting of the pathway. We can refer to the bifurcation as a catastrophe or a sin-gularity. The portal or door to the singularity may well be characterized by chaos oruncertainty, since the path can veer off in many possible directions, as suggested by thebifurcating cones in Figures 3 and 4. At every level that we consider, the bifurcationsof the evolutionary trajectory seem to be locally characterized by chaotic domains.I suggest that these domains are the result of different coalitional possibilities. Thefact that the trajectories can become indeterminate suggests that this may enhance theexploration of the fitness landscape.

A more general remark concerns the role of climate change. Climate has exhibitedchaotic or catastrophic behavior in the past.15 There is good reason to believe that hu-man evolution over the last million years can only be understood in terms of “bursts” ofsudden transformations (Nowak 2011) and that language and culture co-evolve throughgroup or coalition selection (Cavallli-Sforza and Feldman 1981). Calvin (2003) sug-gests that our braininess was cause and effect of the rapid exploration of the fitnesslandscape in response to climatic forcing. For example Figure 2 in the earlier paper(Schofield 2011) showed the rapid changes in temperature over the last 100,000 years.It was only in the last period of stable temperature, the “holocene,” during the last10,000 years that agriculture was possible. One danger of the current climate change

15 Indeed as I understand the dynamical models, the chaotic episodes are due to the complex interactionsof dynamical processes in the oceans, on the land, in weather, and in the heavens. These are very likeinterlinked coalitions of non-gradient vector fields.


N. Schofield

du1(x)du3(x)

du2(x)

H12(x)H23(x)

H3(x)

H1(x)

H2(x)

Figure 4. The failure of half-openness of a preference field

is not just the possibility of a rise in temperature but that climate itself could becomechaotic, destroying the possibility of agriculture and causing the collapse of our civi-lization.

Stringer (2012) calls the theory of rapid evolution during a period of chaotic climatechange “the Social Brain hypothesis.” The cave art of Chauvet, in France dating backabout 36,000 years suggests that belief in the supernatural played an important part inhuman evolution. Indeed, we might speculate that the part of our mind that enhancestechnological/mathematical development and that part that facilitates social/religiousbelief are in conflict with each other (this is suggested by Kahneman 2011). We mightalso speculate that market behavior is largely driven by what Keynes termed specula-tion, namely the largely irrational changes of mood (Casti 2010). Figure 1 in the earlierpaper (Schofield 2011) gave an illustration of the swings in the U.S. stock market overthe last 80 years. This figure certainly suggested that the stock market does not tend toequilibriate.

7. A Moral Compass

If we accept that moral and religious beliefs are as important as rational calculationsin determining the choices of society, then depending on models of preference aggre-gation will not suffice in helping us to make decisions over how to deal with climatechange. Instead, I suggest a moral compass, derived from current inferences madeabout the nature of the evolution of intelligence on our planetary home. The anthropicprinciple reasons that the fundamental constants of nature are very precisely tuned so



that the universe contains matter and that galaxies and stars live long enough to allowfor the creation of carbon, oxygen etc, all necessary for the evolution of life itself.16

Gribbin (2011) goes further and points out that not only is the sun unusual in havingthe characteristics of a structurally stable system of planets, but the earth is fortunate inbeing protected by Jupiter from chaotic bombardment but the Moon also stabilizes ourplanet’s orbit.17 In essence Gribbin gives good reasons to believe that our planet maywell be the only planet in the galaxy that sustains intelligent life. If this is true then wehave a moral obligation to act as guardians of our planetary home. Parfit (2011) argues

“What matters most is that we rich people give up some of our luxuries,ceasing to overheat the Earth’s atmosphere, and taking care of this planetin other ways, so that it continues to support intelligent life. If we are theonly rational animals in the Universe, it matters even more whether weshall have descendants during the billions of years in which that would bepossible. Some of our descendants might live lives and create worlds that,though failing to justify past suffering, would give us all, including thosewho suffered, reason to be glad that the Universe exists.” (Parfit 2011,p. 419)

8. Collapse

In the aftermath of the Great Recession, many authors have argued that the institu-tions that served the west as it industrialized are no longer effective (Ferguson 2012;Oreskes and Conway 2014). An earlier argument by Tainter and Renfrew (1988) onthe basis of a review of the anthropological and archeological literature on the collapseof complex societies is that all such societies develop increasing complex institutions,and that complexity itself induces increasing marginal cost. Without any doubt theinstitutions of capitalism have become more complex over time. Such complexity canbe seen in the Limits to Growth models of Meadows et al. (1972, 2012). If we regard“complexity” as the “rules of the game” then it is certainly plausible that the beha-vior of such a game will be located at the edge of chaos, as suggested above and thussubject to catastrophe. The logic of this theory is that we face the collapse of theAmerican hegemony, with the end of the period of cheap energy and resources. It isalso possible that China will become the new hegemon. China has been able to growrapidly, benefitting from the positive marginal advantages of western economic institu-tions. Although China faces many problems associated with population, urbanizationand environmental degradation, there are also indications that it is concerned to deviseentirely new institutions that may help it to continue to prosper. Indeed it is possiblethat the continued development of China will usher in a completely new world order

16 As Smolin (2007) points out, the anthropic principle has been adopted because of the experimental evi-dence that the expansion of the universe is accelerating. Indeed it has led to the hypothesis that there is aninfinity of universes all with different laws. An alternative inference is the the principle of intelligent design.My own inference is that we require a teleology as proposed in the conclusion.17 The work by Poincare in the late 19th century focussed on the structural stability of the solar system andwas the first to conceive of the notion of chaos.


N. Schofield

that will be entirely different from the system of nation-states that developed from thepost-enlightenment dominance of the West. It has also been suggested that the agricul-tural revolution that occurred at the beginning of the Holocene was accompanied byan ideological revolution associated with a belief in our ability to manipulate Naturefor our own ends (Seddon 2014). This ideology can be seen as a precursor to enlight-enment beliefs. Perhaps we need a new system of morality based on post-Holocenevirtues appropriate to the age we live in rather than to enlightenment “rationality.”

9. Conclusion

Even if we believe that markets are well behaved, there is no reason to infer that mar-kets are able to reflect the social costs of the externalities associated with productionand consumption. Indeed Gore (2006) argues that the globalized market place, whathe calls Earth Inc has the power and inclination to maintain business as usual. If thisis so, then climate change will undoubtedly have dramatic adverse effects, not least onthe less developed countries of the world.18

In principle we may be able to rely on a version of the Jury Theorem (Rae 1960;Schofield 1972a,b; Sunstein 2009), which asserts that majority rule provides an op-timal procedure for making collective choices under uncertainty. However, for theoperation of what Madison called a “fit choice” it will be necessary to overcome theentrenched power of capital. Although we now disregard Marx’s attempt at construct-ing a teleology of economic and political development,19 we are in need of a morecomplex over-arching and evolutionary theory of political economy, embodying a sys-tem of morality that will go beyond the notion of equilibrium and might help us dealwith the future.20

Acknowledgment This paper is based on research supported by NSF grant 0715929.

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